The Multistep Spectral Collocation Methods For Nonlinear Volterra Integral Equations, Volterra Integral Equations With Delays And Volterra Functional Integro-differential Equations With Delays

As an important numerical method, spectral method often provides exceedingly accurate nu?merical results with relatively less degree of freedoms, and has been widely used for scientificcomputations. On the other hand, Volterra integral equations (VIEs), Volterra integral equationswith delays (DVIEs) and Volterra functional integro-differential equations (VFIDEs) are the mod?els of evolutionary problems with memory arising in many applications, such as physical andbiological phenomena, lasers and population growth, etc.. In recent years, some numerical ap?proaches for such problems have become increasingly popular.Among the existing works, numerical methods based on the single-step spectral schemes havebeen frequently used for various linear VoltciTa-type equations. However, this kind of algorithmsis not suitable for long time simulations. Therefore, it is also very important for us to study theniullislep spcclral methods of the nonlinear Vol terra-type equations.The main purpose of this dissertation is to investigate the multistep Legendre-Gauss spec?tral collocation ineihods for nonlinear VIEs, nonlinear VIEs with vanishing variable delays andnonlinear VFIDEs with vanishing variable delays. As theoretical results we fully analyze andcharacterize the/i/j-convergence of the methods under reasonable assumptions. Numerical ex?periments demonstrate that the suggested methods possess the spectral accuracy. In particular,they are very appropriate for various problems with highly oscillating solutions and sleep gradientsolutions, as well as numerical simulations of long time behaviors.This dissertation consists of the following four parts.Firstly, we briefly review the research progress on numerical methods of VIEs, DVIEs andVFIDEs.Secondly, we introduce a multistep Legendre-Gauss spectral collocation method for the non?linear VIEs of the second kind. This method is easy to implement and possesses the high orderaccuracy. In addition, it is very suitable for long time calculations. We also cfcrive the conver?gence of the hp-version of the multistep collocation method under L2-norm. Numerical resultsdemonstrate the spectral accuracy of this approach and coincide well with the theoretical analysis.Thirdly, we propose a multistep Legendre-Gauss spectral collocation method for the nonlinearsecond-kind VIEs with vanishing variable delays. We also provide the convergence analysis of thehp-version of the multistep spectral collocation method under L2-norm. Numerical experimentsconfirm the Lhcorctical expectations.Finally, we develop a multistep Legenclre-Gauss spectral collocation method for【he nonlinearVFIDEs with vanishing variable delays. We also present the convergence analysis of the hp- version of the multistep spectral collocation method under//’-norm. Numerical results are givento demonstrate the high accuracy of our scheme.